SPIE Proceedings [SPIE SPIE Defense, Security, and Sensing - Baltimore, Maryland, USA (Monday 29 April 2013)] Atmospheric Propagation X - Sparse spectrum model for the turbulent phase

Sparse spectrum model for the turbulent phase simulations Mikhail Charnotskii*a

a*Zel Technologies, LLC and NOAA/Earth System Research Laboratory Research Laboratory, 325 Broadway R/PSD-99, Boulder, CO, USA

ABSTRACT

Monte-Carlo simulation of phase front perturbations by atmospheric turbulence finds numerous applications for design and modeling of the adaptive optics systems, laser beams propagation simulations, and evaluating the performance of the various optical systems operating in the open air environment. Accurate generation of two-dimensional random fields of turbulent phase is complicated by the enormous diversity of scales that can reach five orders in magnitude in each coordinate. In addition there is a need for generation of the long “ribbons” of turbulent phase that are used to represent the time evolution of the wave front. This makes it unfeasible to use the standard discrete Fourier transform-based technique as a basis for the Monte-Carlo simulation algorithm. We propose a novel concept for turbulent phase – the Sparse Spectrum (SS) random field. The principle assumption of the SS model is that each realization of the random field has a discrete random spectral support. Statistics of the random amplitudes and wave vectors of the SS model are arranged to provide the required spectral and correlation properties of the random field. The SS-based Monte-Carlo model offers substantial reduction of computer costs for simulation of the wide-band random fields and processes, and is capable of generating long aperiodic phase “ribbons”. We report the results of model trials that determine the number of sparse components, and the range of wavenumbers that is necessary to accurately reproduce the random field with a power-law spectrum.

Keywords: Turbulence simulation, phase screens, Monte-Carlo, Strehl number

1. INTRODUCTION Simulated turbulent phase screens are widely used in imaging, adaptive optics, beam propagation and, optical communication. A variety of simulation techniques has been developed for more than 30 years, but there is still no satisfactory solution that combines the high speed and accuracy. The objective of this paper is to present an efficient Monte-Carlo model that generates 2-D random phase samples ( )rS v

with a structure function that is close to

( ) ( ) ( )[ ] 21,2

<<⎟⎟⎠

⎞⎜⎜⎝

⎛=−+= α

α

CrrRSrRSrD

rrrr. (1)

for a certain range of separations Lrl ≤≤ using the novel Sparse Spectrum (SS) random field concept.

The phase structure function is related to the spatial spectrum of the phase fluctuations ( )KSr

Φ as 1

( ) ( ) ( )[ ]rKKKdrD Srrrr⋅−Φ= ∫∫ cos12 2 . (2)

and this equation is valid for the random fields with stationary increments when the variance is not necessary finite.

A large number of papers 2-33 discussed various methods of the random phase simulation starting from mid 1970s up until recently. A detailed discussion of the techniques described in these papers can be found in 34.

*[email protected]

Atmospheric Propagation X, edited by Linda M. Wasiczko Thomas, Earl J. Spillar, Proc. of SPIE Vol. 8732, 873208 · © 2013 SPIE · CCC code: 0277-786X/13/$18 · doi: 10.1117/12.2016437

Proc. of SPIE Vol. 8732 873208-1

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2. SPARSE SPECTRUM MODEL DESCRIPTION 2.1 Sparse-spectrum concept

Similar to the conventional, FFT-based model (8), the Sparse Spectrum model presents the 2-D random field of phase ( )rSr

as a sum of harmonics

( ) ( )rKiarS n

N

nn

rrr⋅= ∑

=

expRe1

(3)

with random normally distributed complex amplitudes Nnan ,...,2,1, = , having the following statistics

mnnmnmnn saaaaa δ=== *,0,0 (4)

However, in contrast to the FFT model, the wave vectors nKr

are random vectors with probability distributions ( )Kpnr

{ } ( ) KdKpKdKKP nnrrrrr

=+∈ . (5)

Structure function ( )rDr

can be calculated from the SS model equations (3 – 5) as follows:

( ) ( ) ( )[ ]{ }

( )[ ]{ }

( ) ( )[ ].cos1cos11

2

1,

2 ∑∫∫∑==

⋅−=⋅−=−+=N

nnn

K

N

nnn

KarKKpsKdrKsRSrRSrD

nnn

rrrrrrrrr

rr (6)

Clearly, the SS model structure function (14) will match the prescribed structure function (2) when

( ) ( )KKpsN

nnn

rrΦ=∑

=

21

(7)

Assuming that compliance with a prescribed structure function is the only requirement to be satisfied, Eq. (15) is the only constraint on the weights ns and probability distributions ( )Kpn

r. The SS model was first proposed in 35 for sea

surface elevations modeling, and the existence of the spectral sparsity of the sea surface was experimentally validated in 36. Note that the conventional FFT phase screen model can still be presented in the SS form (11) when probability distributions ( )Kpn

r are delta-functions supported at the nodes of the rectangular grid.

2.2 Sparse-spectrum Monte-Carlo model for turbulent phase

The Monte-Carlo model, as presented here, is designed specifically for the power-law structure functions (1), but can easily be modified for any desired spectrum, correlation, or structure function. Writing the spectrum corresponding to the structure function (1) as

( ) ( ) αα −−=Φ 2KBKr

(8)

we use Eq. (2) to relate ( )αB to the coherence radius Cr as

( ) α

α

απ

ααα −

−

⎟⎠⎞

⎜⎝⎛ −Γ

⎟⎠⎞

⎜⎝⎛ +Γ

= CrB

21

212 2

(9)

Based on the isotropy of the phase fluctuations, we choose to use the wave vectors PDFs ( )Kpnr

in the polar coordinate form



( ) ( ) ( )πϕϕϕ

2,2 ddkKpkdkdKpKdKp nnn ==

r (10)

Equation (10) assumes that wave vectors’ directions are uniformly distributed on [ ]ππ ,− in compliance with the statistical isotropy of the phase. Principal equation of the SS model (7) now reduces to

( ) ( ) ααπ −−

=

=∑ 1

1

4 KBKpsN

nnn (11)

There are many different ways to satisfy Eq. (11), and each will result in different Monte-Carlo models for generation of the phase samples with the same structure function (1). We chose a computationally simple model that uses non-overlapping PDFs of the wavenumbers. In addition, given the power law shape of the ( )KS

rΦ and the associated wide

range of the wavenumbers, we rewrite the Eq. (11) in terms of the ( )Kln≡χ , where K is measured in m-1:

( ) ( ) αχαπχ −

=

=∑ eBpsN

nnn 4

1

(12)

Here we used the obvious PDFs relationship ( ) ( )KdKKpdp =χχ . Given the required range of the χ - values

MAXMIN χχχ ≤≤ , we divide it into N segments, and assume that ( )χnp is supported only on the n -th segment

nn χχχ ≤≤−1 . In this case Eq. (12) reduces to N equations

( ) ( ) NneBps nnnn ,...,2,1,,4 1 =≤≤= −− χχχαπχ αχ (13)

With the expectation that the number of segments will be at least several hundreds, and the changes of the right-hand part of (13) inside each interval are negligible, we replace it by a constant on each subinterval

( ) ( ) nnnn

nn eee χχχχχα

αχαχαχ ≤≤−−

≈ −−−

−

− −1

1,1

1 (14)

This choice of constant warrants that the power in the spectral range is conserved. Equations (12 - 14) uniquely determine the PDFs ( )χnp and the weights ns as

( ) ( ) ( ) .1,141

1

nnnn peeBs nn

χχχ

ααπ αχαχ

−=−=

+

−− + (15)

For the numerical trials described in the next Section, we used the log wavenumber subintervals of equal length:

( ) NnNn

MINMAXMINn ,...,2,1,0, =−+= χχχχ (16)

Note that some specific features of the suggested model, such as the non-overlapping intervals (13), uniform PDF approximation (14), and constant subinterval width (15), are nonessential simplifications, and can be avoided at some additional computational costs.

3. NUMERICAL TRIALS For numerical tests of the SS-based Monte-Carlo model, we chose a fixed range of separations Lrl ≤≤ where we want our samples of ( )rS r to be compliant with the statistics prescribed by Eq. (1). The coherence radius was kept constant at

cm10=Cr , but this is not critical, since it is essentially a phase “amplitude” scaling factor rather than a spatial scale for these trials. Note also that scales l and L are not the classical inner and outer turbulence scales, but it is necessary to have 0ll > and 0LL < in order for Eq. (1) to be a valid model for the structure function. The spectral range bounds

MINχ and MAXχ , and the number of spectral components, N, are the undetermined parameters of the model that have to



be estimated in order to create a practical SS-based simulation algorithm and will be determined based on the tests described below.

3.1 Spectral exponent test

The first test is intended to show that the SS concept works for a range of the α - values. We intentionally used a very large number of 000,25=N components, a very wide range of scales from 10-5 m to 104 m and averaged 30,000 samples for each α to show that the SS model generates samples with a structure function given by Eq. (1). Figure 1A shows that for the two orders of magnitude of the separation range, we have a very accurate reproduction of the target structure functions (1). It is noticeable, however, that the accuracy somewhat decays for the largest 611=α . Figure 1B shows the ratio of the structure functions obtained by averaging of the MC model samples to the target structure function (1). Again the 611=α case shows noticeable deviations that are increasing for larger scales. This can be attributed to the redistribution of the energy to larger scales for α approaching the critical value of 2, when samples are represented by tip and tilts only 37. As a result even the four orders of magnitude “padding” at larger scales are not sufficient to capture the contribution of the large scales for the largest 611=α .

Figure 1. A: SS model structure functions for the various exponents α of the power-law (1). Also shown are the best-fit power-law approximations. B: Ratios of the SS model structure functions to the target structure functions (1) for the various exponents α of the power-law (1) obtained from the SS Monte-Carlo model.

3.2 Spectral Range Test

For further tests we chose mm1=l and m1=L , and used 35=α only. The objective of the next series of trials is to determine the range MAXMIN χχχ ≤≤ that needs to be used in the model to accurately reproduce the structure function for separations range Lrl ≤≤ . We present the results in the form of the structure functions ratio, already used in Fig. 1B, and use more intuitive linear scales ( )MAXMINL χπ −≡ exp2 and ( )MINMAXL χπ −≡ exp2 . We used 1000=Ncomponents model and averaged 3000 samples for each point in this test.

Figure 2A shows the sensitivity of the SS structure functions to the choice of the largest SS scale MAXL in terms of the ratios of the SS model – derived structure function to the target structure function (1) for 35=α . The narrowest (1 mm – 1 m) range of scales used matches the l and L values, and shows a very poor reproduction of the target ( )rD . The series of the dashed curves shows the effect of the increasing MAXL causing the better match to the target ( )rD , and terminating with the heavy solid curve corresponding to the (1 mm – 3·103 m) range that provide acceptable 10% deviations from the target ( )rD . Figure 2B shows the dependence of the SS structure function on the smallest scale

y = 14.73x1.17

y = 21.43x1.33

y = 31.48x1.50

y = 45.25x1.66

y = 59.89x1.81

0.01

0.1

1

10

100

0.01 0.1 1

Stru

ctur

e fu

nctio

ns

r, m

1.17

1.33

1.50

1.67

1.83

0.85

0.9

0.95

1

1.05

0.01 0.1 1r, m

1.17

1.33

1.50

1.67

1.83A B



included in the model. Comparison of the thin solid and short dashed curves in Figure 2 shows that expanding the range of scales on the small-scale MINL outside l has a very little effect. The dot-dashed line shows the drastic undesirable effect of using MINL larger than l . Heavy solid line in Figure 4 is the model with acceptable mmLMIN 1= and

mLMAX 3000= is shown as a reference.

Figure 2. A: Dependence of the structure functions on the largest scale included in the SS model MAXL . B: Dependence of

the structure functions on the smallest scale included in the SS model MINL .

Figure 3A shows the dependence of the SS model on the target structure functions ratio for the SS model using a 1 mm – 1000 m range of scales, and various numbers of the components. Even as few as 50 components deliver acceptable accuracy. The overshoot for 20=N is a result of the poor approximation provided by 20 log-uniformly distributed components at the six orders of magnitude spectral range. It is also clear that for 100≥N the accuracy is determined by the finite scale range and not by the number of components. Figure 3B shows similar results for the wider, 1 mm – 3000 m scales range. It is noteworthy that the accuracy of the 20 components model is slightly degraded in comparison to the previous case, owing to the wider spectral range to be represented with insufficient number of components. The accuracy of the models with 501000 ≥≥ N improves as a result of the wider scales range, but there is a wider spread between the 50 and 1000 components models. This suggests that this at least 100 components are needed to represent the 6.5 decimal orders scale range.

Matching of the wavenumbers PDFs to the actual shape of the ( )KSr

Φ most likely would rectify the structure function mismatch for smaller number of components, but will carry additional computation costs. More important, however, is that for small N individual phase samples will show pronounced periodic features not expected for the “natural” phase fluctuations.

Based on these calculations, we conclude that SS models with about 300 components provide an adequate representation of the phase fluctuations with a 310=lL ratio. Note that the FFT – based model would require 4·106 components to capture this range of scales, and the output will be periodic with a 1 m period.

3.3 Number of Components Test

The number N of spectral components does not essentially affect the accuracy of the SS model. In fact it is possible to satisfy the fundamental equation (7) by using a single mode in each sample, provided that the probability distribution of the wave vectors ( )Kp

r has the same shape as the spectrum ( )K

rΦ . While each sample would look as a sinusoid, the

desired spectrum will be properly reproduced for the average over a large set of samples. For our specific Monte-Carlo

0.6

0.7

0.8

0.9

1.00.001 0.01 0.1 1

r, m

1mm - 3000m

1mm - 1000m

1mm - 100m

1mm - 10m

1mm - 1m0.6

0.7

0.8

0.9

1.00.001 0.01 0.1 1

r, m

1mm - 3000m

0.1mm - 10m

0.1mm - 1m

10mm - 1000m

B A



1.10

1.05

1.00

0.95

0.90

N= 1000

- - N= 300

- N= 100

N= 50

N= 20

02 04

\

0;6 0;8

ty ....

,....,...,,,,

a.-... -.....

r, m

1.10

1.05

1.00

0.95

0.90

N. 1000

- - N. 300

0 2 0 4 ---- N= 100 g :

N. 50

.,N. 20

-... -..\-=-...

r, m

model we use a set of the log-uniform probability distributions (15) in order to simplify the computations. As a result the smooth spectral density is approximated by a piecewise constant function (in log scale). Clearly, the larger number of components provides better approximation to the smooth spectrum Eq. (8). The objective of the following test is to determine the acceptable number of components.

We use two ranges of the model scales MINL and MAXL : (1 mm – 103 m) and (1 mm – 3·103 m) to determine the minimum number of spectral components that deliver the samples with acceptable structure functions in the 1 mm – 1 m separation range.

Figure 3. A: Dependence of the SS model to the target structure functions ratio for 1 mm – 1000 m range of scales, and various numbers of the components N. B: Same as Fig. A, but for the 1 mm – 3000 m range of scales.

3.4 Phase Ribbon Test

One of the advantages of the SS model is that it provides non-periodic samples. This property can be used to generate long “ribbons” of phase samples when the spatial extent in one dimension, say x , is much larger than the other, y dimension based on the same set of amplitudes and wave vectors. We generated phase “ribbons” that are 1000 m long along the x-axis and 2 m wide in the y direction. Figure 4 shows the phase structure function for the 1 x 5 m2 rectangle area for N = 1000 and m3000=MAXL accumulated from 2000 independent samples, and referenced at the “ribbon” center. Due to the presence of the very large scales in the SS model, there is no evidence of the structure function saturation along the “ribbon.” Figure 5 shows a sample of the phase surface generated by the model. The presence of the large-scale trend and small scale local variations reflects the rich spectral content of the power-law model (1). Figure 6A shows samples of the phase values along the y direction referenced at the phase at m1=y at the different positions along the “ribbon.” The chart shows random changes of the phase along the “ribbon” for distances up to 1 km. Figure 6B shows phase structure functions calculated based on the 3000 samples similar to the ones shown in Fig. 6A. As expected, structure functions are identical and closely follow the target ( )rD , Eq. (1).

3.5 Strehl Number Test

The optical field at the image plane of a point source viewed through the aperture with the amplitude transparency function ( )rA r and phase distortions ( )rS r is given by

( ) ( ) ( )[ ]riSrrikrArdCru IMIMvvvvv

+⋅−= ∫∫ exp2 (17)

A B



650

600

550

500

450

400

350

300

250

200

150

100

50

0

Figure 4. Same SS structure function for 1 x 5 m2 rectangle area.

where we use the angular coordinate in the image plane, and use shorthand notation C for all non-essential factors. The instantaneous Strehl number for this imaging model can be written as

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −−⎟

⎠⎞

⎜⎝⎛ +⎟

⎠⎞

⎜⎝⎛ −⎟

⎠⎞

⎜⎝⎛ +

Σ= ∫∫ ρρρρρ

rrrrrrrr

21

21exp

21

211# 22

2 RiSRiSRARAdRdSt (18)

where ( )rrAd r∫∫=Σ 2 is the aperture area. Assuming that the phase distortions are a zero-average Gaussian field, we can

calculate the average Strehl number as

( )⎥⎦⎤

⎢⎣⎡−⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −⎟

⎠⎞

⎜⎝⎛ +

Σ= ∫∫∫∫ ρρρρ

rrrrrDRARARddSt

21exp

21

211# 22

2 (19)

For the power-law model phase (1) and a circular “top hat” aperture with radius a, the average Strehl number is a function of a single parameter ( )aD 2 . For ( ) 12 >>aD as follows from (19):

( )[ ] ααα

2222221# −+

⎟⎠⎞

⎜⎝⎛ +Γ≈ aDSt (20)

The dashed and dotted curves in Figure 7A show an average of 104 random Strehl numbers (for each point) calculated according to Eq. (18) using the SS phase samples based on the MC models with various numbers of components. The heavy solid curve shows direct numeric integration of Eq. (19), and the light solid curve – asymptote (20). Well-averaged MC results closely overlap and are undistinguishable on the chart. They are also very close to the analytical results for all, but the largest ( )aD 2 . This should be expected since for m1=a used in these calculations the smallest coherence radius mm1=Cr was equal to the model fine scale l .

While our MC model accurately reproduces the shape of the structure function, the probability distribution of the SS phase is essentially non-Gaussian 35, 36. In particular, this means that the averaging of Eq. (18) does not lead to Eq. (19). However, test results shown in Fig. 7A suggest that the SS phase probability is close enough to Gaussian, at least for the average Strehl number calculations.

For imaging through turbulence, the short-exposure Strehl number is a random variable 38, 39. Our MC model provides the instantaneous values of the Strehl number that can be used to calculate the statistical moments. Under the Gaussian phase assumption, the integral representations for the higher moments of the Strehl number can be readily derived, but



practical calculations of the associated multi-fold integrals can be a difficult task 39. Figure 7B shows estimates of the Strehl number scintillation index (SI) for different numbers of components N in the SS model as dashed, dot-dashed, and solid lines. The dotted curves show some results of the numeric integration of the eight-fold integral formulation of the scintillation index based on the Gaussian statistics of phase (see 39 for details of the numerical calculations). All SS – based curves group very tightly for the lesser ( )aD 2 values and tend to the expected unity value for ( ) 12 >>aD . Strong deviation of the SI for 20=N is not surprising given our earlier observations of the poor structure function representation in this case. There is a noticeable trend of the SI dependence on N near maxima that can be attributed to the different degree of the non–Gaussian behavior for the SS models with a different number of components. Independent numerical integration results are close to the SS–derived SI, and both can benefit from the refinement of the computation techniques.

Figure 5. A sample of the phase surface generated by the model over a 1 x 5 m2 rectangle area.

4. CONCLUSIONS • The Sparse Spectrum model of the random phase offers a very efficient modeling tool that allows representation of a wide range of scales without using the very large number of Fourier components required by conventional FFT.

• The Sparse Spectrum model generates non-periodic samples of practically unlimited size, while preserving desired local statistics at the suggested scale range.

• Essentially non-Gaussian statistics of the Sparse Spectrum phase do not seem to affect the statistics of the Strehl number significantly. This is probably due to the effect of the central limit theorem on the Sparse Spectrum fields with large numbers of components.

• The properly designed Sparse Spectrum models with different number of components, and FFT–based models can generate the random samples with identical first and second moments. Obviously, the full statistics (e.g. higher moments, multipoint PDFs, etc.) of these random fields are different. It is not clear at this point which model is closer to the reality of the turbulence phase, and what are the statistical metrics that can gauge the quality of these different models.

• The specific choices of the non-overlapping and log-uniform segments used here for the wavenumbers PDFs are arbitrary, and mostly are convenience driven. There is still a great potential for the refinement and optimization of the model in the framework of the Sparse Spectrum concept.

• The Sparse Spectrum is computationally simpler than the latest hybrid models using combinations of statistical interpolation, midpoint displacement, spectrum partition, and sub-harmonics.



511

40

;11

20

10

0

-- 10- 100

1000

Target D(r)

05 1

y, m

1.5 2

Figure 6. A: Samples of the phase values along the y direction referenced at the phase at m1=y at the different positions along the “ribbon.” B: Structure functions calculated based on the 3000 samples at different positions along the “ribbon.”

• The Sparse Spectrum is ideally suited for representation of the slowly varying phase statistics. This can be easily modeled by introduction of the time varying parameters ns in Eq. (7). The time scales for these variations can be different for different spectral ranges of the SS model.

• The main disadvantage of the Sparse Spectrum is the inability to use an efficient FFT algorithm. However in many cases it can be compensated by the drastically reduced number of the sparse components required in contrast to the FFT case.

Figure 7. A: Strehl numbers calculated according to Eq. (18) using the SS phase with various numbers of components. Also shown are the results of numerical integration of Eq. (29), and asymptote Eq. (30). B: Strehl number scintillation index for SS model and numerical integrations.

-6

-4

-2

0

2

4

6

0 0.5 1 1.5 2Phas

e

r, m

0 10

100 1000

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05

Stre

hl n

umbe

r

D(2a)

N= 1000

N= 300

N= 100

N= 50

N= 20

Num.

Asym.

0

1

2

3

1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05

Scin

t. In

dex

D(2a)

N= 1000

N= 300

N= 100

N= 50

N= 20

A

A B

B



• The non-Gaussian probability distribution of the phase can be of concern, but it is also present in many FFT-based methods that use the phase-only random coefficients. Moreover, non-Gaussian probability of the SS field is asymptotically Gaussian for a large number of components, as our Strehl number calculations showed.

• The Sparse Spectrum model produces non-ergodic samples, but this does not seem to be an issue for the modeling community, since the conventional FFT–based samples are also not ergodic.

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Proc. of SPIE Vol. 8732 873208-10


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SPIE Proceedings [SPIE SPIE Defense, Security, and Sensing - Baltimore, Maryland, USA (Monday 29 April 2013)] Atmospheric Propagation X - Sparse spectrum model for the turbulent phase